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This module is always available. It provides access to the mathematical
functions defined by the C standard.

These functions cannot be used with complex numbers; use the functions of the
same name from the cmath module if you require support for complex
numbers. The distinction between functions which support complex numbers and
those which don’t is made since most users do not want to learn quite as much
mathematics as required to understand complex numbers. Receiving an exception
instead of a complex result allows earlier detection of the unexpected complex
number used as a parameter, so that the programmer can determine how and why it
was generated in the first place.

The following functions are provided by this module. Except when explicitly
noted otherwise, all return values are floats.

Return fmod(x,y), as defined by the platform C library. Note that the
Python expression x%y may not return the same result. The intent of the C
standard is that fmod(x,y) be exactly (mathematically; to infinite
precision) equal to x-n*y for some integer n such that the result has
the same sign as x and magnitude less than abs(y). Python’s x%y
returns a result with the sign of y instead, and may not be exactly computable
for float arguments. For example, fmod(-1e-100,1e100) is -1e-100, but
the result of Python’s -1e-100%1e100 is 1e100-1e-100, which cannot be
represented exactly as a float, and rounds to the surprising 1e100. For
this reason, function fmod() is generally preferred when working with
floats, while Python’s x%y is preferred when working with integers.

Return the mantissa and exponent of x as the pair (m,e). m is a float
and e is an integer such that x==m*2**e exactly. If x is zero,
returns (0.0,0), otherwise 0.5<=abs(m)<1. This is used to “pick
apart” the internal representation of a float in a portable way.

The algorithm’s accuracy depends on IEEE-754 arithmetic guarantees and the
typical case where the rounding mode is half-even. On some non-Windows
builds, the underlying C library uses extended precision addition and may
occasionally double-round an intermediate sum causing it to be off in its
least significant bit.

Return True if the values a and b are close to each other and
False otherwise.

Whether or not two values are considered close is determined according to
given absolute and relative tolerances.

rel_tol is the relative tolerance – it is the maximum allowed difference
between a and b, relative to the larger absolute value of a or b.
For example, to set a tolerance of 5%, pass rel_tol=0.05. The default
tolerance is 1e-09, which assures that the two values are the same
within about 9 decimal digits. rel_tol must be greater than zero.

abs_tol is the minimum absolute tolerance – useful for comparisons near
zero. abs_tol must be at least zero.

If no errors occur, the result will be:
abs(a-b)<=max(rel_tol*max(abs(a),abs(b)),abs_tol).

The IEEE 754 special values of NaN, inf, and -inf will be
handled according to IEEE rules. Specifically, NaN is not considered
close to any other value, including NaN. inf and -inf are only
considered close to themselves.

Return the IEEE 754-style remainder of x with respect to y. For
finite x and finite nonzero y, this is the difference x-n*y,
where n is the closest integer to the exact value of the quotient x/y. If x/y is exactly halfway between two consecutive integers, the
nearest even integer is used for n. The remainder r=remainder(x,y) thus always satisfies abs(r)<=0.5*abs(y).

Special cases follow IEEE 754: in particular, remainder(x,math.inf) is
x for any finite x, and remainder(x,0) and
remainder(math.inf,x) raise ValueError for any non-NaN x.
If the result of the remainder operation is zero, that zero will have
the same sign as x.

On platforms using IEEE 754 binary floating-point, the result of this
operation is always exactly representable: no rounding error is introduced.

Note that frexp() and modf() have a different call/return pattern
than their C equivalents: they take a single argument and return a pair of
values, rather than returning their second return value through an ‘output
parameter’ (there is no such thing in Python).

For the ceil(), floor(), and modf() functions, note that all
floating-point numbers of sufficiently large magnitude are exact integers.
Python floats typically carry no more than 53 bits of precision (the same as the
platform C double type), in which case any float x with abs(x)>=2**52
necessarily has no fractional bits.

Return e raised to the power x, minus 1. Here e is the base of natural
logarithms. For small floats x, the subtraction in exp(x)-1
can result in a significant loss of precision; the expm1()
function provides a way to compute this quantity to full precision:

>>> frommathimportexp,expm1>>> exp(1e-5)-1# gives result accurate to 11 places1.0000050000069649e-05>>> expm1(1e-5)# result accurate to full precision1.0000050000166668e-05

Return x raised to the power y. Exceptional cases follow
Annex ‘F’ of the C99 standard as far as possible. In particular,
pow(1.0,x) and pow(x,0.0) always return 1.0, even
when x is a zero or a NaN. If both x and y are finite,
x is negative, and y is not an integer then pow(x,y)
is undefined, and raises ValueError.

Unlike the built-in ** operator, math.pow() converts both
its arguments to type float. Use ** or the built-in
pow() function for computing exact integer powers.

Return atan(y/x), in radians. The result is between -pi and pi.
The vector in the plane from the origin to point (x,y) makes this angle
with the positive X axis. The point of atan2() is that the signs of both
inputs are known to it, so it can compute the correct quadrant for the angle.
For example, atan(1) and atan2(1,1) are both pi/4, but atan2(-1,-1) is -3*pi/4.

The mathematical constant τ = 6.283185…, to available precision.
Tau is a circle constant equal to 2π, the ratio of a circle’s circumference to
its radius. To learn more about Tau, check out Vi Hart’s video Pi is (still)
Wrong, and start celebrating
Tau day by eating twice as much pie!

A floating-point “not a number” (NaN) value. Equivalent to the output of
float('nan').

New in version 3.5.

CPython implementation detail: The math module consists mostly of thin wrappers around the platform C
math library functions. Behavior in exceptional cases follows Annex F of
the C99 standard where appropriate. The current implementation will raise
ValueError for invalid operations like sqrt(-1.0) or log(0.0)
(where C99 Annex F recommends signaling invalid operation or divide-by-zero),
and OverflowError for results that overflow (for example,
exp(1000.0)). A NaN will not be returned from any of the functions
above unless one or more of the input arguments was a NaN; in that case,
most functions will return a NaN, but (again following C99 Annex F) there
are some exceptions to this rule, for example pow(float('nan'),0.0) or
hypot(float('nan'),float('inf')).

Note that Python makes no effort to distinguish signaling NaNs from
quiet NaNs, and behavior for signaling NaNs remains unspecified.
Typical behavior is to treat all NaNs as though they were quiet.